Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation
Multi-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order $2N$ in terms of trigonometric functions is deduced. All these solutions dep...
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| Format: | Article |
| Language: | English |
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Emrah Evren KARA
2021-12-01
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| Series: | Universal Journal of Mathematics and Applications |
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| Online Access: | https://dergipark.org.tr/tr/download/article-file/1909237 |
| _version_ | 1797350195934003200 |
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| author | Pierre Gaillard |
| author_facet | Pierre Gaillard |
| author_sort | Pierre Gaillard |
| collection | DOAJ |
| description | Multi-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order $2N$ in terms of trigonometric functions is deduced. All these solutions depend on $2N-1$ real parameters. A third representation in terms of a quotient of two real polynomials depending on $2N-2$ real parameters is given; the numerator is a polynomial of degree $2N(N+1)-2$ in $x$, $y$ and $t$ and the denominator is a polynomial of degree $2N(N+1)$ in $x$, $y$ and $t$. The maximum absolute value is equal to $2(2N+1)^{2}-2$. We explicitly construct the expressions for the first third orders and we study the patterns of their absolute value in the plane $(x,y)$ and their evolution according to time and parameters.\\ It is relevant to emphasize that all these families of solutions are real and non singular. |
| first_indexed | 2024-03-08T12:41:15Z |
| format | Article |
| id | doaj.art-4ed51402d720491982b17c9d5e27a38d |
| institution | Directory Open Access Journal |
| issn | 2619-9653 |
| language | English |
| last_indexed | 2024-03-08T12:41:15Z |
| publishDate | 2021-12-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Universal Journal of Mathematics and Applications |
| spelling | doaj.art-4ed51402d720491982b17c9d5e27a38d2024-01-21T09:29:21ZengEmrah Evren KARAUniversal Journal of Mathematics and Applications2619-96532021-12-014415416310.32323/ujma.9788751225Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I EquationPierre Gaillard0Université de BourgogneMulti-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order $2N$ in terms of trigonometric functions is deduced. All these solutions depend on $2N-1$ real parameters. A third representation in terms of a quotient of two real polynomials depending on $2N-2$ real parameters is given; the numerator is a polynomial of degree $2N(N+1)-2$ in $x$, $y$ and $t$ and the denominator is a polynomial of degree $2N(N+1)$ in $x$, $y$ and $t$. The maximum absolute value is equal to $2(2N+1)^{2}-2$. We explicitly construct the expressions for the first third orders and we study the patterns of their absolute value in the plane $(x,y)$ and their evolution according to time and parameters.\\ It is relevant to emphasize that all these families of solutions are real and non singular.https://dergipark.org.tr/tr/download/article-file/1909237kadomtsev petviasvili eqautionfredholm determinantswronskiansrational solutions |
| spellingShingle | Pierre Gaillard Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation Universal Journal of Mathematics and Applications kadomtsev petviasvili eqaution fredholm determinants wronskians rational solutions |
| title | Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation |
| title_full | Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation |
| title_fullStr | Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation |
| title_full_unstemmed | Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation |
| title_short | Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation |
| title_sort | multi parametric families of real and non singular solutions of the kadomtsev petviasvili i equation |
| topic | kadomtsev petviasvili eqaution fredholm determinants wronskians rational solutions |
| url | https://dergipark.org.tr/tr/download/article-file/1909237 |
| work_keys_str_mv | AT pierregaillard multiparametricfamiliesofrealandnonsingularsolutionsofthekadomtsevpetviasviliiequation |